An Experimental Test of Precautionary Bidding

Transcription

1 An Experimental Test of Precautionary Bidding Martin G. Kocher Department of Economics, University of Munich, Germany Department of Economics, University of Gothenburg, Sweden Julius Pahlke Department of Economics, University of Munich, Germany Stefan T. Trautmann * Department of Economics, Tilburg University, The Netherlands January, 2012 Abstract: Auctions often involve goods exhibiting a common knowledge ex-post risk that leads to precautionary bidding with DARA bidders in expected utility models, i.e., bidders reduce their bids by more than the appropriate risk premium. Because the degree of riskiness of the good and bidders risk aversion are difficult to observe in field settings, we conduct experimental auctions that allow identifying the precautionary premium directly. Our results provide clear support for the precautionary bidding conjecture. Bidders are better off when a risky object rather than an equally valued sure object is auctioned. This result is robust when controlling for potentially confounding decision biases. JEL classification: C91, D44, D81 Keywords: precautionary bidding, prudence, auction, experiment * We are grateful for very helpful comments by Louis Eeckhoudt, Dan Levin, Johannes Maier, Theo Offerman, Harris Schlesinger, Larry Samuelson, and Matthias Sutter, and for valuable feedback at numerous occasions, including seminars in Linz, Maastricht, Munich and Tilburg, the ESA European Regional Meeting in Innsbruck 2009, the European Workshop on Experimental and Behavioral Economics (EWEBE) 2009 in Barcelona, the North American Winter Meeting of the Econometric Society at the ASSA Meeting in Atlanta 2010, and the European Economic Association Meeting in Glasgow Financial support from the Munich Experimental Laboratory for Economic and Social Sciences (MELESSA), the LMU-Ideenfonds at the University of Munich, and from a NWO-VENI grant to Stefan Trautmann is gratefully acknowledged. Corresponding author: Stefan Trautmann, Dept. of Economics & CentER, Tilburg University, P.O. Box 90153, 5000LE Tilburg, The Netherlands, phone: , fax: , s.t.trautmann@uvt.nl 1

2 1. Introduction Consider an auction with pure ex-post risk: The value of the auctioned good is risky, with the risk being independent of private or common value components and signals thereof. The risk is known ex-ante and is common knowledge among buyers. In the language of decision theory, the auctioned good is a risky lottery. Esö and White (2004) theoretically study auctions with ex-post risk in the affiliated value model by Milgrom and Weber (1982). They show for the standard first-price auction that bidders exhibiting decreasing absolute risk aversion (DARA) unambiguously reduce their bids by more than the appropriate risk premium, an effect they call precautionary bidding. 1 The intuition is that DARA bidders prefer higher income in the case they win the auction and have to bear the ex-post risk involved in the good, and therefore bid more conservatively. This effect is similar to the precautionary saving motive where agents transfer current wealth into future periods with more income uncertainty (Kimball (1990)). Examples for auctions with ex-post risk are numerous and financially significant. Television rights for Olympic Games are usually auctioned off before the host city is selected from a set of competitors. The winner bears the risk of a more or less attractive host, a risk arising from information unavailable to any bidder at the time the rights are allocated. In procurement auctions, unpredicted events that affect production costs add ex-post risk to the winning bidder s profit. More generally, all goods whose resale value or quality is uncertain ex-ante to all buyers involve some ex-post risk. Precautionary bidding, if empirically relevant, has several important implications for auction design in general, and more specifically for information revelation by sellers and information acquisition by buyers. For instance, according to theory, sellers have an incentive to reduce the risk exhibited by the auctioned object as much as possible, and buyers have a strategic incentive to disregard some information. Because of its importance for the prediction of auction outcomes and for economic design, it is essential to study the empirical validity of precautionary bidding. Despite the widespread occurrence of ex-post risk in auctions and its relevance, no empirical study on precautionary bidding has been conducted so far. A direct measurement of precautionary bidding with field data is not easily available, because it requires the independent observation of both the bidders risk tolerance and the riskiness of the good. In 1 In their article, they provide results for various auction formats. We focus on first-price auctions here. 2

3 order to provide the first empirical assessment of precautionary bidding, we conduct experimental laboratory auctions where the controlled setting allows us to identify and quantify the precautionary premium directly. Our main experiment finds strong support for bidding behavior being consistent with precautionary bidding in first-price auctions. Bidders are significantly better off in first-price auctions when a risky object rather than an equally valued sure object is auctioned. Although our empirical hypotheses build on Esö and White s (2004) theoretical framework, the experimental tests that we conduct for the data from the experiment are in fact model-free, relying only on observable certainty equivalents. In addition, we provide results for a parametric expected utility analysis. The latter shows that risk-averse Nash equilibrium predicts bidding behavior in deterministic auctions reasonably well, but it fails to predict bidding behavior in auctions with ex-post risk. This strongly corroborates the finding from our model-free analysis. In a control experiment, we address alternative explanations based on behavioral biases in decision-making that might have influenced our results in the main experiment. The data from this control experiment are in line with our conclusions from the main experiment and show the robustness of the precautionary bidding effect. 2 The remainder of the paper is laid out as follows. In the next section we introduce the theoretical framework and the predictions of the precautionary bidding model. In Section 3 we present our experimental design in detail. Section 4 reports the results from the main experiment, and Section 5 provides evidence from the control experiment and performs additional robustness checks. In Section 6 we discuss our results and conclude. 2. Theoretical framework and predictions Almost all analyses of bidding behavior in auctions today assume that objects are non-risky, although agents are often assumed to have noisy signals regarding the true value of the object. Risk aversion plays a role in first-price auctions, because it reduces bid shading and, therefore, increases the bid in comparison to the risk-neutral Nash equilibrium (e.g., Cox, Smith and Walker, 1982, 1985; Maskin and Riley, 1984; Kagel, Harstad and Levin, 1987). The only study so far that considers pure ex-post risk in common auctions formats is Esö and 2 Though the main interest of the paper is to study bidding behavior, to our best knowledge the current study more generally provides the first experimental evidence for prudence effects in a microeconomic optimization setting (in contrast to direct preference elicitation, as in Baillon (2011), Deck and Schlesiger (2009), Ebert and Wiesen (2011), or Noussair et al. (2011)). 3

4 White s (2004) theoretical analysis of the affiliated value model by Milgrom and Weber (1982). We follow their setup and assume that there are n potential bidders for a given object. Bidder i receives a private signal s i [ s, s]. The joint distribution of the signal follows the properties of affiliation described in Milgrom and Weber (1982). The risky ex-post monetary value of the object for bidder i is v argument, i v( s, s ) z, where v is strictly increasing in its first i i i s i denotes the highest signal of all bidders other than bidder i, and z i is the realization of a random variable, z~ i, with zero mean. The realizations of the random variable come from a symmetric joint distribution and are independent of the signals, but they can be (perfectly) correlated across bidders. By definition, if z~ i is non-degenerate, the object is risky. The utility function u is strictly concave and thrice differentiable. For DARA bidders, 2 2 ( u / x ) /( u / x) is strictly decreasing. Esö and White (2004) prove in this framework that, holding everything else constant, DARA bidders in the first-price auction have unambiguously higher indirect utilities in a symmetric equilibrium when z i is non-degenerate, i.e., the object is risky. 3 In the following, we give a brief intuition for the result and derive empirical predictions, building on Esö and White s hypothesis, regarding the buyer s equilibrium bids for a risky good and for her (riskfree) certainty equivalent of this risky good. Our identification of precautionary bidding in the experiment will be based on the comparison of bids for risky lotteries and their certainty equivalents on the individual level. In the first-price auction, for risk-averse agents who maximize expected utility with a DARA utility function, the introduction of a mean-preserving ex-post risk has three effects. First, the value of the object for the agent is reduced by the risk premium. Agents replace the value of the risky object v i by its certainty equivalent CE i (v i ) before making their bids. For risk-averse bidders by definition CE v ) E[ v ], where E [.] denotes the expected value. i ( i i Second, the riskiness of the object introduces a background risk, making bidders become more risk-averse regarding other risks (Eeckhoudt, Gollier and Schlesinger, 1996). Hence, in the presence of ex-post risk they will shade their bids less than predicted by the appropriate 3 The main result extends immediately to situations where another independent noise is added to make already noisy valuations even riskier (Kihlstrom, Romer and Williams (1981)). 4

5 risk premium, because the possibility of reducing the chance to lose the object in the auction becomes more attractive than the risky gain of a higher payoff by decreasing the bid. There is a third effect, however, the precautionary effect. This effect causes bidders to bid less aggressively because each extra unit of income is more valuable to them in the case they win the auction for the risky object as opposed to its certainty equivalent, due to the background risk. In other words, increasing the probability of winning the auction through a higher bid becomes more costly in the presence of ex-post risk. This effect is related to the prudence premium (Gollier 2001; Eeckhoudt and Schlesinger, 2006). Taking all three effects together, Esö and White (2004) prove that for DARA bidders in equilibrium the total effect of ex-post risk on the reduction of one s bid is unambiguously larger than just the risk premium. 4 Our empirical strategy to identify precautionary bidding is based on the comparison of bids for risky objects and bids for their certainty equivalents. By construction, both goods are equally valuable. If bids for risky objects were larger than bids for their certainty equivalents, the effect of background risk on bid shading would dominate, in contrast to the theory. If bids for risky objects were smaller than for their certainty equivalents, however, the precautionary effect must dominate, in accordance with the theory. Hence, we can formulate our main hypothesis. HYPOTHESIS 1: In the affiliated private value first-price auction, buyers bids b i for a risky object will be lower than their bids for a risk-free object whose value is equal to their individual certainty equivalent of the risky object, i.e., b( v) b( CE ( v)). (1) i i i i i The precautionary bidding effect is similar to the precautionary saving motive where agents transfer more wealth into the future when they face a higher future income risk (Kimball, 1990). Compared to precautionary saving, however, in first-price auctions individual risk attitudes and the level of riskiness of the object affect equilibrium bidding through multiple channels that point into different directions (see effects two and three above). In particular, increased risk aversion leads to less bid-shading (effect two). This makes the result of an unambiguously negative effect of precautionary bidding on the bid the more remarkable. It also implies, ceteris paribus, that buyers are better off bidding for a risky 4 For DARA bidders the prudence premium is larger than the risk premium. 5

6 object than for the equivalent risk-free object, and that in equilibrium, they will anticipate this effect and hold lower beliefs. As noted before, in most settings the (perceived) riskiness of the good and the degree of bidders risk aversion cannot easily be measured independently in the field. Moreover, direct comparisons between bids for risky and risk-free goods with an identical certainty equivalent typically cannot be constructed. Our experimental test of precautionary bidding directly compares bids for independently elicited certainty equivalents with bids for the appropriate risky objects on the individual level. Hypothesis 1 is derived from the precautionary bidding model under the premise of expected utility theory. Since our empirical test is only based on comparisons of bids for risky prospects and observable certainty equivalents, however, it is in fact model-free, and Hypothesis 1 can be interpreted as a behavioral definition of the precautionary effect in firstprice auctions. 3. Experimental design 3.1 The auction In the experiment, we follow the affiliated private value implementation of Kagel et al. (1987), adjusted to the setup in which either sure prospects or risky prospects are sold. In particular, for sure prospects the subjects knew their private valuation, and for risky prospects they knew the prospect they could win in the auction. Their valuations were correlated because of a two-step procedure used to draw valuations and prospects from some interval of the whole payoff range (Kagel et al., 1987, p. 1277). That is, a high private value observed by the agent makes it more likely that the other bidders also have high values. More specifically, our experimental subjects participated in a series of twelve anonymous first-price auctions with 3 bidders. In each auction, subjects received an endowment of 10, which they could use to bid for an object of risky or certain value. In order to produce matched bids for prospects and their individual certainty equivalents, in four of the twelve auctions a participant was bidding for two risky prospects and for her two corresponding individual certainty equivalents, which were elicited before in four out of the twelve auctions (see Section 3.2 for details of the elicitation procedure). We call the specific bidder whose certainty equivalent is used in these 4 auctions the bidder of interest, and the matched observations for the risky object and the appropriate individual certainty equivalent 6

7 an auction pair. In the remaining eight auctions the private valuation of the bidder was drawn according to the procedure described in the following paragraph, with each of the other two bidders being the bidder of interest four times in turn. 5 For risk-free prospects, the bidder of interest s certainty equivalent for the matched prospect provided her private valuation v i. For the other two bidders in the group the valuations were determined as follows: v i was first reduced by a random number z (-), which was drawn from the interval [ 0, 2], and then increased by a random number z (+) from the D same interval. The number obtained, v0 vi z( ) z( ), forms the midpoint of a 4-interval in which all three deterministic valuations (hence, superscript D) lie. Subjects did not learn the midpoint of the interval. Hence, the valuations of the other two bidders within a group D D were drawn from the interval [ v 0 2, v0 2]. By construction, the value v i lies in the interval and can assume any position in this interval, like the two other valuations. Figure 1 illustrates the procedure for v i = 4.21, v 0 = 5.11, and Bidder 1 as bidder of interest. For risky prospects, the procedure was similar. The bidder of interest had to bid for a risky prospect, presented in terms of its expected value plus or minus a fixed and announced amount K {2,3,4 } with equal probability. 6 For each prospect, K follows uniquely from the rewriting of the gamble presentation in the preceding risk elicitation stage of the experiment (see Section 3.2 for details). The risky prospects for the other two bidders were determined as described for sure objects, but taking as v i the expected value of the risky prospect for the bidder of interest. This gives expected values for the other two bidders in the range R R [ v 2, v0 0 2], to which the same ex-post risk of size K was added for all bidders. FIGURE 1: ILLUSTRATION OF VALUATION ASSIGNMENT z (-)=0.8 Bidder 1: 4.21 z (+)=1.7 Bidder 3: 6.07 Bidder 2: Note that no auction outcomes were revealed until all twelve bids had been made by the subjects to avoid learning, wealth, or reputation effects. 6 The presentation is identical to the theoretical formulation of ex-post risk as a noise added to a private valuation. Note that K took on different values for different auctions, but was always clearly announced before bidding. 7

8 For example, assume that Bidder 1 indicated a certainty equivalent of 4.21 for the prospect [0.5: 6.36, 0.5: 2.36] in the risk elicitation task preceding the auction experiment. In the auction experiment, she would face one auction as illustrated in Figure 1 and another auction that would be described as offering an object with risky value of 4.36 that will, with a probability of 0.5 each, either be increased or decreased by an amount of K = 2. The two other bidders in the group would face a randomly drawn sure valuation out of the permissible range in the first auction and a randomly drawn risky valuation out of the permissible range (according to the procedure described above) in the second auction. Subjects were instructed about the general procedure of drawing values and the method of affiliation in great detail (see the Appendix B for the experimental instructions), but they were neither aware of the presence of a bidder of interest, nor of the fact that we took their certainty equivalents and prospects from the preceding risk elicitation stage, nor of the private valuations of the two other bidders. More precisely, they were simply told that the private valuations of the three bidders come from a 4-interval lying within a larger interval and were distributed randomly along this 4-interval (which was true by construction). Everything else was common knowledge among participants. Neither intermediate auction results nor bids were revealed before the end of the experiments, and groups stayed together for all twelve auctions. 7 After the twelve auctions, one auction was randomly selected for real payment of the full amount in euro. The auction winners paid their bids and received the payoff from the auction, possibly depending on the result of the ex-post risk, and they kept the rest of their 10-endowment that they had not used for bidding. Subjects who did not win the auction kept their endowment of 10. All randomizations concerning risky equal-chance events in the experiment were conducted by throwing dice at the subjects desks. Remember that the twelve auction rounds give us, per subject, two matched auction pairs (bids) for identical sure and risky valuations, and four more observations for bids for sure valuations. That is, in total we know individual valuations in eight out of the twelve auctions and can use this information for econometric analyses. We do not observe the valuation for subjects who were not the bidder of interest in the remaining four auctions for risky prospects. Only the bidder of interest submits a bid for a prospect for which we 7 Groups were formed randomly, but subjects who were close to each other in their risk attitude rank (within a session of 15 subjects) from the preceding risk elicitation had a higher chance to end up in the same group. This procedure approximates the assumption of identical risk attitudes for bidders in EW s model and was explained in neutral terms to the participants (see the instructions). 8

9 previously elicited her valuation in the risk preference elicitation stage that is described in the next sub-section. 3.2 Elicitation of risk preferences At the beginning of the experiment, we elicited subjects certainty equivalents for eleven risky prospects. All prospects were binary-outcome prospects with a 50% chance of each outcome (see Table 1, columns 1-3). Certainty equivalents were elicited using the Becker- DeGroot-Marschak (1963) incentive mechanism (henceforth, BDM). Subjects were asked to state for each prospect their minimum selling price p s between the low and the high outcome of the prospect. They knew that a random buying price p b between these two outcomes would be drawn to determine if the prospect is sold to the experimenter if p p, in which case the subject received the randomly drawn buying price, or is not sold otherwise, in which case the subject received the outcome of the prospect. The BDM mechanism has been used extensively in preference elicitation and is valid in our expected utility framework (e.g., Karni and Safra, 1987; Halevy, 2007). However, no BDM randomizations or risky prospects were resolved at this stage in order to prevent wealth differences between subjects in the auctions to come. Subjects were instructed that at the end of the experiment they would be paid on the basis of the outcome of one of the risky prospects or receive the random buying price, depending on the outcome of the BDM procedure. Prospect no. High payoff (prob. 50%) TABLE 1: RISKY PROSPECTS USED IN THE EXPERIMENT Low payoff (prob. 50%) Expected value Average CE with BDM a b s Average CE with choice list b Numbers in columns 2-6 show amounts in. a CE = Certainty equivalent; BDM = Becker-DeGroot-Marschak mechanism. b For an explanation of the right-most column, see Section 5. 9

10 3.3 Laboratory protocol and subjects Computer-based experiments were conducted at the experimental laboratory MELESSA of the University of Munich, using the experimental software z-tree (Fischbacher, 2007) and the organizational software Orsee (Greiner, 2004). Seventy-five undergraduate students without experience in auction experiments participated in 5 sessions with 15 subjects each. Sessions lasted up to two hours, and the average final payoff was 23.75, including a showup fee of 4. Subjects received written instructions which were read aloud and had the opportunity to ask questions in private. Examples and/or test questions were given for each stage of the experiment, and a stage only began when all subjects correctly understood the procedures. The experiment started with the risk elicitation stage. Subjects received instructions for this stage, but knew that there would be further stages in the experiment. Upon completion, subjects received instructions for the second stage of the experiment. This stage was purely instructional, i.e., it was intended to make subjects acquainted with bidding in auctions with and without ex-post risk. Subjects participated in twelve affiliated private value first-price sealed-bid auctions for six risky and six sure prospects. Auctions were held anonymously in groups of three people, with new groups formed in every auction. Again, subjects with a similar risk rank from the elicitation stage had a higher chance to end up in the same group in each auction. Subjects could bid from an endowment of 10 tokens in each auction. All bids within a group, the winning bid and the winner were announced immediately after each auction to acquaint subjects with the affiliated value model and with bidding for uncertain prospects. Only at the end of the entire experiment, one auction from this training stage was randomly selected and subjects were paid according to the outcome. Furthermore, payments were scaled down by a factor of 1/10 (compared to the main auction experiment in stage 3). With the exception of the size of earnings and the specific procedure of taking prospects and certainty equivalents from the risk elicitation stage, this second stage of the experiment was identical to the main auction stage that was to follow. After stage 2 had ended, subjects received instructions for the main stage of the experiment, the twelve auctions as described in Section 3.1. Remember that subjects were not informed about the matched structure and the bidder of interest in the main stage of the experiment. Note that, while prospects were identical to the ones in the risk elicitation for the bidder of interest, the presentation of the prospects was quite different. They were framed in terms of ex-post risk instead of binary gambles and were 10

11 not easily recognized as the same prospects as in the first stage 8, also because the training stage introduced a significant time lag. At the end of the experiment, all random aspects of the experiment were resolved, and subjects learned what they had earned in each of the three stages of the experiment. They were paid privately and in cash and then dismissed from the laboratory. 4. Results of the main experiment Column 5 in Table 1 shows the average certainty equivalents for the lotteries elicited in the first stage of the experiment. On average, subjects exhibit risk aversion, with CEs being smaller than expected values for all prospects (Wilcoxon signed-ranks tests; p < 0.01). Figure 2 provides clear evidence consistent with the hypothesized precautionary bidding effect, b( v) b( CE ( v)). In Panel A, for the 150 matched auction pairs (2 auction pairs for each of i i i i i the 75 subjects), we show the number of pairs in which a buyer made a lower, identical, or higher bid for the risky prospect than for its CE. Clearly, risky prospects elicit lower bids than their certainty equivalents from the same individual (Wilcoxon signed-ranks test; p < 0.01). In 109 matched auction pairs, a lower bid was submitted for the risky prospect than for its certainty equivalent, in comparison to only 35 pairs with higher bids for the risky prospect. There were virtually no identical bids, suggesting that subjects did not simply remember prospects and their certainty equivalents from stage 1 and tried to be consistent. 9 Panel B of Figure 2 shows that for risky prospects the whole distribution of bids shifts toward the left compared with bids for the matched certainty equivalents. Increased bid shading for risky prospects is also consistent within-person, with 50 subjects always shading more for risky, 14 always shading less, and only 11 with mixed behavior. Alternatively, one can perform a parametric utility analysis to assess precautionary bidding. The first benchmark measure for the evaluation of sure and risky prospects in our experiment are risk-neutral Nash equilibrium bids (Kagel et al. 1987), given in equation (2) n b( vi ) vi exp[ ( vi ( s ))] (2) n n ( n 1) 2 8 If so, it would make our results even stronger. Details are provided in the next section. 9 Note that even if some bidders of interest had recognized the prospects from the risk elicitation stage in the later auction stage and understood the construction of our matched auction pairs, they had no more relevant information on values and intervals than the other bidders in the group. 11

12 In the case of our experiment, n = 3 (number of bidders), ε = 2 (radius of the smaller interval), and s 0 (lower bound of the larger interval). For each auction and each bidder we calculate the risk-neutral Nash equilibrium bids from individual valuations. We find significant overbidding for the sure prospects, consistent with previous findings in the experimental literature for risk-averse subjects (Cox et al., 1988). 10 The actual bids are significantly higher than the risk-neutral Nash bids (Wilcoxon signed-ranks test, p < 0.01). For risky prospects we find significant underbidding with respect to the risk-neutral Nash equilibrium bids (Wicoxon signed-ranks test, p < 0.01). As a robustness check, one can aggregate relative overbidding and underbidding on the individual level. The basic result for the 75 subjects does not change, and both tests are still significant (p < 0.01 for sure prospects, and p < 0.05 for risky prospects). FIGURE 2: COMPARISON OF BIDS FOR RISKY PROSPECTS A WITH BIDS FOR THEIR CERTAINTY EQUIVALENTS B Notes: Panel A: Number of pairs in which the bid for a risky prospect was (lower/identical/higher) than the bid for its certainty equivalent (within-person comparisons). Panel B: Distribution of bids (in ) for risky prospects and for their deterministic certainty equivalents; % of subjects. In a next step, we estimate an individual utility function for each subject based on the data we have from the certainty equivalent elicitation stage (see Table 1). This allows us to calculate risk-averse Nash equilibrium bids for each bidder and auction based on the 10 Overbidding is a common empirical phenomenon in first-price auctions. Explanations fall roughly into three categories: risk aversion, inter-personal comparisons, and non-equilibrium behavior or learning. Surveys of the literature are, for instance, provided in Crawford and Iriberri (2007) and Engelbrecht-Wiggans and Katok (2007).

13 individual risk aversion parameters and the (expected) value of the risky or deterministic prospects. These Nash bids are then again compared to the actual bids. More specifically, we use nonlinear least squares estimations to fit a constant relative 1 r risk aversion utility function (CRRA), u( x) x /(1 r), with risk aversion parameter r, for all 75 subjects individually. 11 For each risky and for each risk-free auction, we can then calculate risk-averse Nash equilibrium bids and compare them to the actual bids. Nash bids are calculated according to the equilibrium bidding formula in Kagel et al. (1987). b( v, ) v i i 2 2 (1 ) 1 (1 ) 2 exp[ n [ vi n n ( n 1 ) n ( s )]/ 2 (1 )], (3) with = r (risk parameter of the utility function). Equation (3) can only be applied if r < 1. Several subjects in our experiment are more risk-averse than that. We therefore truncate their risk aversion parameter r at 0.99, which underestimates the actual level of their risk aversion. Nash bids are virtually identical to actual bids for sure prospects (Wilcoxon signed-ranks test, p = 0.83). However, we observe strong underbidding compared to the benchmark solution for risky prospects (Wilcoxon signed-ranks test, p < 0.01). The robustness check of using relative bids, aggregated on the individual level, gives the same general picture (p = 0.55 for sure prospects; p < 0.01 for risky prospects). In order to avoid arbitrary parametric utility assumptions and to fully exploit the modelfree nature of our test of precautionary bidding, we estimate the quantitative effect of ex-post risk on bids by using regression analyses, controlling for the panel structure with eight observations per subject. 12 Plotting valuations and bids suggested a linear specification. 13 We include a dummy variable for bids made for risky prospects and a coefficient that captures the interaction of valuations with the presence of the ex-post risk. Model I in Table 2 shows that for sure prospects buyers shade their bids by 15 cents per euro valuation. In the presence of risk, bids are reduced by another 18 cents per euro valuation for the prospect. That is, the precautionary bidding effect is observed, because equally valuable risky and sure prospects elicit significantly different bids. Bidders shade their bid approximately twice as much if the good is affected by ex-post risk than when it is not. 11 We selected a CRRA utility function because it has been widely applied in the literature on risk aversion and first-price auctions. It obviously has the DARA property. 12 Remember that for each subject, we know private valuations for two risky and six sure prospects. 13 Models with non-linear specifications as a robustness check are provided in Appendix A. 13

14 TABLE 2: DETERMINANTS OF BIDDING BEHAVIOR (FIXED EFFECT PANEL REGRESSION) Dependent variable: I II III IV Bid (BDM) (BDM, excl. (BDM, excl. (BDM) bids with low risk rank a ) bidders with low risk rank b ) Valuation 0.849** (0.024) 0.874** (0.026) 0.831** (0.026) 0.854** (0.024) Risk (0.281) (0.301) (0.330) Risk Valuation ** ** ** - (0.045) (0.051) (0.051) Risk size K c (0.092) Risk size K Valuation * (0.014) Constant (0.155) (0.163) (0.178) (0.154) # Obs. (bids) # Obs. (bidders) R Risk rank: Discrete variable ranging from 1 (least risk-averse) to 15 (most risk-averse). a Bids with lowest risk ranks excluded (ranks 1 to 3 out of 15). b Bidders with lowest risk rank in their session excluded (ranks 1 to 3 out of 15). c Risk size K: Margin between lowest and highest outcome of the prospect ( 2, 3 or 4).**/* significant at 1%/5% level; BDM = Becker-DeGroot-Marschak mechanism. Standard errors in parenthesis. In models II and III in Table 2 we address the robustness of the effect with respect to the risk-aversion rank of a specific certainty equivalent or of a specific bidder. The least riskaverse certainty equivalent for a given prospect has rank 1 and the most risk-averse certainty equivalent has rank 15 within an experimental session. Similarly, the subject in each session with the lowest average risk-aversion rank over all eleven prospects has rank 1 etc. From a psychological perspective, it could be argued that the effect found in the auction is driven by subjects who do not exhibit a stable risk attitude or who reveal too large certainty equivalents by mistake in the first stage of the experiment, and successively make very low bids in the auction (regression to the mean). Notice that buyers who provided relatively high certainty equivalents will have low risk ranks. We distinguish between individual certainty equivalents that are high for a certain prospect and may come from different bidders for different prospects (model II) and bidders who generally state high certainty equivalents (model III). In Table 2, we show the regression results when we exclude observations or bidders with the lowest three risk ranks. In the two alternative specifications the precautionary effect stays both economically and statistically significant. Standard errors increase due to the loss of more than 100 observations in each model, but the estimates are very robust. Note that while 14

15 we have chosen to exclude the lowest three ranks, our results do not change when we exclude fewer or some more of the high certainty equivalents. Further, a standard regression-to-themean explanation would equivalently imply that low-certainty-equivalent bidders should have higher actual valuations, and, therefore, increase their bids for the risky prospects compared to the matched certainty equivalents. This effect, were it present, would reduce the observed precautionary bidding effect. The right-most column in Table 2 considers the comparative static effect of increases in risk. Because the size of the ex-post risk varied among lotteries (remember that K {2,3,4} ), we can test whether the bid-shading effect is correlated with the size of the risk. Indeed, we observe that the median increase in bid shading for risky prospects over the bid shading for the paired CEs equals 0.20, 1.21, and 1.30 for risks of size 2, 3, and 4, respectively. That is, an increase in bid shading is obtained for increases in ex-post risk. Regression IV in Table 2 confirms the effect. For each euro increase in the risk level K, bid shading increases by about 3 cents per euro valuation. 5. Control experiment The results of the main experiment support the precautionary bidding hypothesis. However, several alternative, though ad-hoc, explanations of our data are conceivable. One could, for instance, claim that all subjects consistently reveal too high certainty equivalents in the elicitation stage. Although the BDM mechanism is widely used for preference elicitation (Halevy, 2007, p. 507), an upward bias for BDM selling prices has sometimes been reported (Isaac and James, 2000; Plott and Zeiler, 2005). If all subjects reported too large certainty equivalents, the negative effect of risk on bids could be explained by downward revision of the valuations of risky prospects in the auctions. Another possible, non-expected-utility explanation builds on the behavioral concept of loss aversion. If outcomes are described in terms of gains and losses from some reference point, subjects hold lower valuations of a prospect compared to a description in terms of gains only. In the risk elicitation stage of the experiment, prospects were described as binary gambles with two positive outcomes. Because of the affiliated value structure with sure and risky prospects, it was more natural to describe prospects in terms of a valuation plus ex-post risk in the auctions. The natural ex-post risk description, however, may have led subjects to frame the prospects in terms of an equal-chance gain or loss from the reference point of the 15

16 sure valuation. This might have made the risky prospects less attractive than in the binary presentation in the risk elicitation, and, therefore, appear less valuable than the matched certainty equivalents. Although these behavioral biases provide more ad-hoc explanations than the precautionary bidding model, they have been shown to be descriptively relevant in other situations and may provide a psychologically convincing alternative explanation to the equilibrium model. We therefore conducted a control experiment that is able to assess potential effects of a selling price bias and loss aversion. 5.1 Design and hypotheses The control experiment (conducted with 75 new subjects in five sessions) was completely identical to the main experiment, except for the following two features. First, the subjects certainty equivalents were elicited by a choice list. For each prospect, subjects made 21 choices between the risky prospect and a sure payoff, with all choices shown simultaneously on the screen (see screenshot in Appendix B). The lowest sure payoff was equal to the low outcome of the prospect, and the highest sure payoff was equal to the high payoff of the prospect, and these two choices were actually pre-determined for the subjects on the screen in order to enforce stochastic dominance. The 19 choices between the high and the low sure payoff were equally spaced in monetary units. These choices had to be filled in by the subjects, and the certainty equivalent was calculated as the midpoint between the highest sure amount for which the subject prefers the risky prospect, HS i, and the lowest sure amount for which she prefers the sure payoff, LS i, i.e. CE i = (HS i + LS i )/2. Because we needed a unique switching point to calculate individual certainty equivalents for the subsequent auction stages of the experiment, we only allowed a single switching point for each individual in the choice list. As in the main experiment, at the end of the experiment one prospect was randomly selected for real pay. For this prospect, one of the 21 choices was randomly selected, and subjects were paid for this stage according to their decisions for the selected choices. The second design change regards the inclusion of another choice list at the end of the experiment that has been interpreted as a measure of loss aversion and has been widely used recently (Fehr and Götte, 2007, p. 316; Gächter, Johnson and Herrmann, 2007; Fehr, Götte and Lienhard, 2008). Subjects are offered a series of prospects, giving an equal chance of either a gain or a loss that they could choose to play or not to play (Table 3). They were free to accept or reject any prospect, that is, we did not require single switching from acceptance 16

17 to rejection as the loss increases along the list. 14 Payments for this choice list were according to decision in all six choices, depending on the outcome of the risky prospects. TABLE 3: CHOICE LIST MEASURE OF LOSS AVERSION Prospect (50% 50%) Accept to play? Lose 2 or win 6 Yes O No O Lose 3 or win 6 Yes O No O Lose 4 or win 6 Yes O No O Lose 5 or win 6 Yes O No O Lose 6 or win 6 Yes O No O Lose 7 or win 6 Yes O No O Adapted from Gächter et al., For losses smaller than 6, rejecting to play the prospect implies a significant loss in expected value that may be explained more easily by a gain-loss framing and a kinked utility function of wealth changes than by a concave utility of wealth. It has also been shown that the predictions of reference-dependent utility models hold mainly for people who reject most of the prospects in this choice list (Fehr and Götte, 2007). While we do not aim to add to the debate regarding utility curvature versus loss aversion, we call subjects who reject more prospects in this task more loss averse, in line with the alternative behavioral hypothesis we aim to test. Assuming the loss-aversion explanation for the choice list clearly implies that the precautionary effect should be driven by the most loss-averse subjects. Loss-averse subjects could value the prospects lower if presented in terms of ex-post risk in the auction rather than as a binary lottery in the initial risk elicitation stage, leading to a reduction of bids for risky prospects compared to their elicited certainty equivalents. This leads to the following two hypotheses originating from behavioral considerations. HYPOTHESIS 2 (BDM Selling): Hypothesis 1 holds only for certainty equivalents elicited through BDM selling prices. HYPOTHESIS 3 (Loss Aversion): Hypothesis 1 holds only for loss averse bidders. 14 In fact, all subjects had a single switching point. 17

18 5.2 Results of the control experiment Table 1 in section 3.2 shows in the right-most column that under the choice list procedure the elicited certainty equivalents were not smaller than under the BDM selling price procedure. In fact, certainty equivalents for prospects 4, 7, 9, and 11 were even significantly larger for the choice list procedure (Mann-Whitney-U-tests, p < 0.05); all other certainty equivalents were not significantly different for the two methods. These first results indicate already that there was no upward bias through the BDM elicitation of certainty equivalents. Further, Figure 3 shows an identical pattern as for the main experiment, with the precautionary effect being even stronger on average. Of the 150 matched pairs of auctions the risky prospect elicited lower bids than its certainty equivalent in the large majority of cases (Wilcoxonsigned-ranks-test, p < 0.01). Again, the whole distribution of bids shifts towards the left, compared with bids for the matched certainty equivalents. We can therefore clearly reject behavioral Hypothesis 2. FIGURE 3: COMPARISON OF BIDS FOR RISKY PROSPECTS WITH BIDS FOR THEIR CERTAINTY A EQUIVALENTS IN THE CONTROL EXPERIMENT B Notes: Panel A: Number of pairs in which the bid for a risky prospect was (lower/identical/higher) than the bid for its certainty equivalent (within-person comparisons). Panel B: Distribution of bids (in ) for risky prospects and for their deterministic certainty equivalents; % of subjects. As before, conducting a parametric utility analysis for risk-free prospects, we observe significant overbidding in comparison to the risk-neutral Nash equilibrium bids (Wilcoxonsigned-ranks-test, p < 0.01), and for risky prospects we observe risk-neutral Nash bids that are much larger than actual bids (Wilcoxon-signed-ranks-test, p < 0.01). Overbidding now does not vanish when one compares actual bids to the risk-averse equilibrium Nash bids

19 based on CRRA utility functions (Wilcoxon signed-ranks test, p < 0.01) for sure prospects. However, underbidding is nevertheless highly significant for risky prospects (Wilcoxon signed-ranks test, p < 0.01). Not surprisingly, given the similarity of the descriptive results from the main experiment and the control experiment, all our conclusions regarding precautionary bidding from the utility analysis for the main experiment remain valid for the choice list procedure. This is also true for taking average bids, aggregated on the individual level, as the basis for the statistical comparison. As in the main experiment we estimate the quantitative effect of ex-post risk on bids using fixed effects panel regressions, shown in Table 4. Model I in the table shows our basic regression, now for the choice list (CL) experiment. The results are very similar to the main experiment, with bid shading of about 15 cents per euro valuation and a significant precautionary effect of another 28 cents per euro reduction under ex-post risk. In models II to IV we test for differences between the precautionary effect jointly for BDM and CL elicitation stages, using the complete data from both experiments. Model II uses all observations and includes an interaction dummy taking up the difference between the BDM and the CL experiment. The precautionary effect remains significant and its magnitude (a 20 cent reduction in the bids per euro valuation) is considerable. TABLE 4: DETERMINANTS OF BIDDING BEHAVIOR (FIXED EFFECT PANEL REGRESSION) - TOTAL Dependent variable: I II III IV Bid (CL) (CL & BDM) (CL, LA dummy) (CL, LA continuous) Valuation 0.853** (0.020) 0.851** (0.016) 0.857** (0.020) 0.856** (0.020) Risk (0.256) (0.190) (0.254) (0.257) Risk valuation ** (0.039) ** (0.032) ** (0.040) ** (0.046) Risk valuation CL * - - (0.017) Risk valuation LAd ** - (0.022) Risk valuation LAc ** (0.008) Constant (0.137) (0.103) (0.136) (0.137) # Obs. (bids) # Obs. (bidders) R **/* significant at 1%/5% level; BDM = Becker-DeGroot-Marschak mechanism; CL = choice list mechanism; LAd = loss aversion dummy, LAc = loss aversion continuous. Standard errors in parenthesis. 19

20 Figure 4 shows the distribution of the number of prospects rejected in the loss aversion measurement task. Note that all subjects switched at most once, and all switched from accepting the first prospects (see Table 3, small losses) to rejecting the later prospects (larger losses). FIGURE 4: LOSS AVERSION # of subjects Number of rejected prospects The median number of rejected prospects is 4, replicating the findings in Fehr et al. (2008) and Gächter et al. (2007). In regression model III we include the loss aversion measure as a dummy for those bidders who reject four or more prospects (median split), and in regression model IV we include the raw number of rejected prospects. Both regressions show that loss aversion does increase the precautionary bidding effect. The loss aversion measures increase the model fit and add significantly to the precautionary effect. However, the coefficients for the precautionary bidding effect stay at around 20 cents per euro valuation and remain highly significant. Thus, our results clearly reject hypothesis 3. The precautionary bidding effect is robust and cannot be explained solely by the two behavioral effects. Finally, the comparative static effect of increases in risk (the level of K) on the degree of bid shading for risky prospects emerges strongly also in the choice-list based experiment, as it did in the main experiment Median increases in bid shading equal 1.00, 1.48, and 1.57 for risk sizes of 2, 3, and 4. A regression analogously to model IV in Table 2 reveals an additional shading of about 6 cents per euro valuation for each euro increase in risk (p < 0.001). 20

21 6. Discussion and conclusion Esö and White (2004) showed theoretically that ex-post risk in affiliated value auctions has an unambiguous effect for bidders with decreasing absolute risk aversion: bids for risky prospects in the first-price auction are discounted by more than the appropriate risk premium. This is a strong result given the several simultaneous effects of risk aversion on bid shading in the first-price auction. If precautionary bidding is descriptively relevant, it has important implications for optimal information collection and revelation by sellers, strategic information acquisition by buyers and, more generally, auction design. An empirical assessment of precautionary motives in auction bidding cannot be obtained easily, however, because it requires independent knowledge of risk and risk attitudes, both difficult to measure precisely in the field. We designed an experimental auction for risky and sure prospects that aims to provide the first empirical assessment of precautionary bidding. Our study directly compared bids for risky prospects with bids for their relevant certainty equivalents on the individual level. It thus allows for a model-free measurement and gives a behavioral definition of the precautionary premium. We find robust evidence that is consistent with the predicted effect. Bids are significantly lower for risky prospects than for the appropriate certainty equivalent for a large majority of our experimental subjects. That is, bidders are significantly better off bidding for a risky object than for an equally valued risk-free object. Consistent with the experimental auction literature, we find on average overbidding with respect to the riskneutral Nash equilibrium for sure objects and that the risk-averse Nash equilibrium under expected utility describes bidding behavior for sure objects reasonably well. Corroborating the precautionary bidding effect, in the presence of ex-post risk, there is significant underbidding with respect to the risk-neutral and the risk-averse Nash equilibrium bids in our data. Although the empirically observed effect in the experiments is consistent with the idea of a precautionary premium, alternative interpretations are possible. Two behavioral interpretations, based on the elicitation of certainty equivalents and on loss aversion, were rejected in the control experiment. Another potential interpretation involves naive beliefs about their competitors bidding behavior (Costa-Gomes and Weizsäcker 2008; Heinemann et al. 2009). Subjects may believe that other subjects shade the bids for risky prospects more strongly than for sure prospects. It is not clear, however, why they would expect a discount that is larger than the expected risk premium. Thus, a belief-based explanation would rather 21

22 depend on subjects systematically overestimating other people s risk premia. The existing literature provides little support for such an effect (Ball et al. 2010; Faro and Rottenstreich 2006). Moreover, in the precautionary bidding model, equilibrium beliefs should also be lower in the presence of ex-post risk, making it impossible to distinguish naive beliefs from equilibrium beliefs about other bidders bids. We therefore think that naively biased believes do provide a convincing alternative explanation for the observed behavior in the auctions. Another concern about the interpretation of our result as a precautionary effect relates to the unobserved degree of prudence. As Kimball (1990, p. 54) argued, the precautionary effect relates to the propensity of people to prepare and forearm oneself on the face of uncertainty. In terms of the current analysis, if a buyer wins the auction and has to carry the risk, she wants to be prepared by holding more wealth and will bid less aggressively for the risky good. Although the precautionary effect is thus quite intuitive in risky auction settings, it may come as a surprise to observe it in a laboratory experiment. The finding is less surprising, however, given that a considerable level of risk aversion in auction experiments is a standard finding. More specifically, decreasing absolute risk aversion (DARA) has also been observed in laboratory experiments. For instance, Levy (1994) conducted a dynamic portfolio choice experiment where subjects made investment decisions under changing wealth levels. Payoffs were given in terms of a few thousand experimental euro, but they translated into typical laboratory payoffs, since the market earnings were divided by a factor of Levy found clear evidence for decreasing absolute risk aversion in terms of experimental wealth. No effect of real wealth on risk taking in the experiment has been observed, however. Levy suggested that subjects make their decision within the frame of payoffs relevant in the experiment and therefore show sensitivity to otherwise rather small changes in payoffs. Similar findings are provided in Deck and Schlesinger (2010) and Noussair et al. (2011), who measure prudence directly using methods in Eeckhoudt and Schlesinger (2006). Both papers show clear evidence of strong prudence for typical experimental stakes, like the ones used in the current paper. Nousair et al. (2011) explicitly test for CARA and reject it in favor of DARA. These findings lend support to the interpretation of our results in terms of precautionary bidding. For bidders with decreasing absolute risk aversion, absolute prudence is larger than absolute risk aversion, leading to the additional precautionary premium. The current paper demonstrates increased bid shading in experimental first price auctions for risky prospects compared to risk free prospects. Thinking of, e.g., auctions for consumer products at online platforms, auctions for art, and auctions for licenses or 22

23 procurement contracts, the typical stakes and risks are much larger in the real world than in our experiment. We therefore expect precautionary bidding to be an important factor, affecting bids and prices, in such non-laboratory markets as well. In some settings, however, precautionary bidding effects may be mitigated by other influences that lead to an upward bias in bidding. Goeree and Offerman (2003) test explanations of the winner s curse using auctions with noisy signals of an uncertain private value. In the context of the current study, this would be similar to resolving the risky prospects ex-ante and providing subjects with a noisy signal of the outcome. Goeree and Offerman observe too optimistic bidding for these private value auctions, similar to the winner s curse for common values, an effect they call the news curse. In situations where buyers receive noisy signals of the value of a risky good, the precautionary effect may therefore be counterbalanced by the news curse. Related settings where ex-post risk may not necessarily lead to precautionary effects involve auctions with resale opportunities (Haile 2003) and license auctions with aftermarkets (Janssen and Karamychev 2009). With potentially countervailing effects on bids, it seems a fruitful direction for future research to study the comparative influence of expost risk on auction outcomes, deriving from different market structures in controlled experimental settings. In a similar vein, within the precautionary bidding framework, a direct empirical test of the predicted market structure effects, including the buyers selection into auctions for risky or risk-free goods and the incentives for sellers to invest in the reduction of ex-post risk, would be desirable. 23

24 References Baillon, Arelien (2011). Prudence under Uncertainty. Working paper, Rotterdam. Ball, S., C. C. Eckel, and M. Heracleous (2010). Risk Aversion and Physical Prowess: Prediction, Choice, and Bias. Journal of Risk and Uncertainty 41, Becker, G. M., M. H. de Groot, and J. Marschak (1963). Stochastic Models of Choice Behavior. Behavioral Science 8, Costa-Gomes, M. and G. Weizsäcker (2008). Stated Beliefs and Play in Normal-Form Games. Review of Economic Studies 75, Cox, J., V. L. Smith and J. M. Walker (1982). Auction Market Theory of Heterogeneous Bidders. Economic Letters 9, Cox, J., V. L. Smith and J. M. Walker (1985). Experimental Development of Sealed-Bid Auction Theory: Calibrating Controls for Risk. American Economic Review 75, Cox, J., V. L. Smith and J. M. Walker (1988). Theory and Individual Behavior of First-Price Auctions. Journal of Risk and Uncertainty 1, Crawford, V. P. and N. Iriberri (2007). Level-k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner s Curse and Overbidding in Private-Value Auctions? Econometrica 75(6), Deck, C. and H. Schlesinger (2010). Exploring Higher Order Risk Effects. Review of Economic Studies 77(4), Ebert, S. and D. Wiesen (2011). Testing for Prudence and Skewness Seeking. Management Science 57, Eeckhoudt, L., C. Gollier, and H. Schlesinger (1996). Changes in Background Risk and Risk Taking Behavior. Econometrica 64, Eeckhoudt, L. and H. Schlesinger (2006). Putting Risk in Its Proper Place. American Economic Review 96, Engelbrecht-Wiggans, R. and E. Katok (2007). Regret in Auctions: Theory and Evidence. Economic Theory 33, Esö, P. and L. White (2004). Precautionary Bidding in Auctions. Econometrica 72, Faro, D. and Y. Rottenstreich (2006). Affect, Empathy, and Regressive Mispredictions of Other's Preferences Under Risk. Management Science 52, Fehr, E. and L. Götte (2007). Do Workers Work More if Wages Are High? Evidence from a Randomized Field Experiment. American Economic Review 97,

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26 Milgrom, P. R. and R. J. Weber (1982). A Theory of Auctions and Competitive Bidding. Econometrica 50, Noussair, C. N., S. T. Trautmann, and G. van de Kuilen (2011). Higher Order Risk Attitudes, Demographics, and Financial Decisions. Working paper, CentER. Plott, C. R. and K. Zeiler (2005). The Willingness to Pay Willingness to Accept Gap, the Endowment Effect, Subject Misconceptions, and Experimental Procedures for Eliciting Valuations. American Economic Review 95,

27 Appendix: Not for publication [will be made available online] Appendix A: Non-linear specifications TABLE A.1: NON LINEAR SPECIFICATIONS (FIXED EFFECT PANEL REGRESSION) Dependent variable I II III Bid (BDM) (CL) (CL, LA dummy) Valuation 1.059** 1.076** 1.089** (Valuation)² (0.009) Risk 1.51 (0.792) Risk Valuation ** (0.123) F(2,520) =635.23** (0.045) F(2,520) =10.44** (0.110) F(2,520) =904.35** * (0.008) (0.743) (0.222) F(2,520) =26.49** (0.108) F(2,518) =939.96** * (0.007) 0.30 (0.732) (0.220) F(2,518)= 5.17** Risk (Valuation)² (0.020) ) (0.016) (0.016) Risk Valuation LAd (0.076) F(2,518)= Risk (Valulation)² LAd * (0.010) 10.13** Constant (0.396) (0.377) * (0.371) # Obs. (bids) # Obs. (bidders) R **/* significant at the 1%/5% level; BDM = Becker-DeGroot-Marschak mechanism.; CL = choice list mechanism. LAd = loss aversion dummy. Standard errors in parenthesis. 27

28 Appendix B: Experimental instructions (translated from German) Welcome to the experiment and thank you for participating! From now on please do not speak with other participants General Procedure The purpose of this experiment is to study decision making. In the experiment you can earn money which will be paid to you in cash after the experiment. During the experiment you and the other participants are requested to make decisions. Your decisions, as well as the decisions of the other participants, will determine your monetary payoff according to the rules explained below. The whole experiment will take about two hours. If you have any questions during the experiment, please raise your hand. One of the experimenters will come to answer your questions at your desk. Anonymity In some parts of the experiment you will be grouped with other participants. Neither during the experiment nor afterwards you or the other participants will learn about the identity of other group members. Neither during the experiment nor afterwards the other participants will learn about your experimental earnings. We will never connect names with experimental results. At the end of the experiment you will have to sign a receipt about your personal earnings which only serves for accounting purposes. The sponsor of this experiment does not receive any experimental data. Auxiliaries You are provided with a pen on your desk. For calculations you will find a link to the Windows calculator on the screen. The Experiment The experiment consists of three parts. You will receive detailed instructions for each part after finishing the previous. In each part you can earn money. The sum of earnings will determine your final income. Part 1 16 Part 1 consists of a sequence of lotteries. Such a lottery could be structured as follows. 16 For the main experiment, i.e., the Becker-deGroot-Marschak mechanism. 28

29 In the above example you would earn 10 with 50% probability and 5 with 50% probability. For each lottery you have two possibilities: 1. you can gamble or 2. you can sell the lottery. Proceedings are as follows: You are asked to state a minimal selling price for the presented lottery. Minimal selling price denotes the price for which you are willing to sell the lottery. This price has to be within a predetermined range. For the above example the range would be from 5 to 10. After stating a minimal selling price (an -amount within the given range with two digits behind the comma) the computer randomly generates a buying offer. The offer is drawn from the same interval which predetermines the range of your choice in the above example, between 5 and 10. Each two-digit number within this interval can be drawn with the same probability. The computer s buying offer is purely random and totally independent from your chosen minimal selling price. Afterwards the computer s buying offer and your chosen minimal selling price will be matched. If the computer s buying offer is higher or equal to your minimal selling price, you sell the lottery to the computer and receive an amount equal to the computer s buying offer. If the computer s buying offer is smaller than your minimal selling price, no sale takes place. You gamble and receive the lottery outcome. The procedure of the gamble will be explained in detail below. Example 1: Let s assume for the lottery shown above you choose a minimal selling price of 7. Let s further assume the computer randomly generates a buying offer of In this case the computer s buying offer is at least as high as your minimal selling price. You sell the lottery to the computer and receive an amount equal to the computer s buying offer, namely Example 2: Let s once more assume you choose a minimal selling price of 7 for the lottery shown above. This time the computer randomly generates a buying offer of Then the computer s buying offer is lower than your minimal selling price. You do not sell the lottery to the computer. You keep the lottery and gamble. Hence, you either receive 5 with 50% probability or 10 with 50% probability. Please note: The randomly generated computer offer is independent of your decision about your minimal selling price. Since in case of a purchase your earnings are not determined by your minimal selling price but 29

30 by the computer s buying offer you should truly state the minimal price for which you are just willing to sell the lottery. Altogether you will state minimal selling prices for 11 lotteries. At the end of the experiment the computer randomly picks one lottery. Since you don t know which one, it is in your own interest to consider all your decisions for all the lotteries carefully. Then, the computer randomly generates a buying offer. If the buying offer is higher or equal to your minimal selling price, you sell the lottery to the computer and receive an amount equal to the buying offer. If the buying offer is smaller than your minimal selling price, no sale takes place. In this case you gamble and receive the outcome of the lottery. More precisely, the experimenter comes to your desk and you roll a six-sided dice. For the example above you would receive 5 if you roll the numbers 1, 2 or 3 or 10 if you roll the numbers 4, 5 or 6. At the top right corner you will find a timer which gives you some temporal orientation for your decision. You can exceed this time limit (especially for the initial decisions, this might most likely be the case). Part 2 17 In part 2 in each round (in each auction) all participants will be matched in groups of three. The group composition may change from auction to auction. However, you will always be matched with participants who have a similar risk attitude. (As a measure of risk attitude we use your decisions of part 1. From now on, none of your decisions will influence subsequent parts of the experiment). You and both other group members will take part in an auction for fictitious goods. For such a good you receive a private valuation (V). This private valuation may deviate from valuations that the two other members of your group receive. Private valuations are determined as follows: In a first step the computer will draw a random number out of a larger interval. Let s assume that the computer randomly chooses This amount subsequently serves as the midpoint of a smaller interval. Later the private valuations will be drawn from this smaller interval. The smaller interval always has a width of four, meaning that in our example your private valuation as well as the private valuations of both other group members will be drawn from an interval between 7.00 and Let s assume the computer randomly allocates you a private valuation of You will learn about your private valuation before the auction starts. In this case you know that this number is drawn from a smaller interval with width 4, and you also know that the midpoint of the smaller interval is drawn from a larger interval. But you do not know the midpoint of the smaller interval. After all group members learned about their private valuations, each group member bids for the good [bid=(b)]. Each group member receives an endowment (E) of 10. Bids above the endowment are allowed. Please note that this may possibly cause losses which will be subtracted from gains from other parts of the experiment. A group s highest bidder acquires the good and pays her bid. Outbid group members do not have to pay their bids. In case of a tie, a coin toss decides. Earnings are determined as follows: Earnings: - Highest bidder: E B + V - Outbid group members: E Some auctioned goods, however, exhibit risk. The risk structure is always the same. With 50% probability your private valuation increases by a certain amount (R) and with 50% probability your private valuation decreases by the same amount. Let s assume an amount (R) of 3. In this case the earnings of the highest bidder will be either reduced or increased by 3 both with a probability of 50%. The amount (R) is identical for all group members (of course, only the winner has to bear the risk). Prior to each auction you will always learn if the auctioned good exhibits risk and if so by which amount (R) the winner s earnings will be increased or reduced. Altogether, you will participate in 12 auctions. Subsequent to each auction you will learn whether or not you have purchased the good. In addition, you learn about the other group members bids. In case the auctioned good exhibit some risk, the resolution of the risk will take place at the end of the experiment. 17 Handed out after completion of Part 1. 30

31 For each of the 12 auctions you have an endowment of 10. At the end of the experiment, one auction will be randomly selected and the results of this auction will be paid out in cash. Since you don t know which one, it is in your own interest to consider all your decisions for all 12 auctions carefully. Each group member receives her earnings from this auction. Since this part is supposed to make you familiar with bidding in an auction and to give you a better understanding of the auction mechanism all earnings will be divided by a factor of 10. Thus, an outbid player in the selected auction receives 10 * 0.1 = 1. A player who submitted the highest bid in the selected auction will receive her endowment minus her bid plus her valuation (if the good exhibits some risk: plus/minus (R)) divided by 10. If in the selected auction you have purchased a good exhibiting a risk, the resolution of the risk takes place at the end of the experiment. More precisely, the experimenter will come to your desk and you roll a six-sided dice. For the numbers 1, 2 or 3 your earnings will be reduced by the amount (R), and for the numbers 4, 5 or 6 your earnings will be increased by the same amount. Part 3 18 This part is very similar to part 2. Again, all participants will be matched in groups of three to participate in a number of auctions. As you already know from part 2, you will always be matched with participants exhibiting a similar risk attitude (as a measure of risk attitude we use again your decisions in part 1). Prior to each auction you will learn about your private valuation which will be determined similarly to part 2. Unlike in part 2, in this part your earnings will NOT be divided by the factor 10. As in part 2, you bid either for goods with a certain value or for goods with a risky value depending on the auction. For each auction you receive an endowment of 10. Bids above the endowment of 10 are allowed, but in case you make a loss, it will be subtracted from gains stemming from other parts of the experiment. Earnings are determined as described in the instructions for part 2. Altogether, you will participate in 12 auctions. Unlike in part 2, subsequently to each auction you will neither learn whether you have purchased the good, nor what others have bid. Instead, after submitting of bids the next auction starts. If the auctioned good exhibit some risk, the resolution of this risk takes place at the end of the experiment. At the end of the experiment one auction will be randomly selected and the results of this auction will be paid out in cash. Since you do not know which one will be chosen, it is in your own interest to consider all your decisions for all 12 auctions carefully. Each group member receives her earnings from this auction. Examples: Example 1 (non-risky good): Players A, B and C have been grouped together. For a non-risky good they receive the following valuations: A: 4.50; B: 8.10; C: 6.50 a) Let s assume knowing their valuations players submit the following bids: A: 4.00; B: 6.00; C: 5.00 Player B submitted the highest bid and thus bought the good. She has to pay a price equal to her bid, namely This results in the following earnings in this auction: A: (E); B: (E) 6.00 (B) (V) = 12.10; C: (E) 18 Handed out after completion of Part 2 31

32 In case this auction is selected to determine payoffs, players A and C receive 10.00, and player receives B b) Let s assume knowing their valuations players submit the following bids: A: 4.00; B: 8.10 ; C: 5.00 Player B submitted the highest bid and thus bought the good. She has to pay a price equal to her bid, namely This results in the following earnings in this auction: A: (E); B: (E) (B) (V) = 10.00; C: In case this auction is selected to determine payoffs, all players receive c) Let s assume knowing their valuations players submit the following bids: A: 3.00; B: 6.00; C: 9.00 Player C submitted the highest bid and thus bought the good. She has to pay a price equal to her bid, namely This results in the following earnings in this auction: A: (E); B: (E); C: (E) (B) (V) = 7.50 In case this auction is selected to determine payoffs, players A and B receive 10.00, and player receives C d) Let s assume knowing their valuations players submit the following bids: A: 15.00; B: 8.00; C: 4.00 Player A submitted the highest bid and thus bought the good. She has to pay a price equal to her bid, namely This results in the following earnings in this auction: A: (E) (B) (V) = ; B: (E); C: (E) In case this auction is selected to determine payoffs, players B and C receive and player A makes a loss of This loss will be deducted from gains made in other parts of the experiment. Example 2 (risky good): Players A, B and C have been grouped together. For a risky good they receive the following valuations: A: 11.70; B: 9.10; C: 8.30 The good exhibits a risk. Its value will increase by 3 (R) with 50% probability or decrease by 3 with 50% probability. a) Let s assume knowing their valuations players submit the following bids: A: 11.00; B: 5.00; C:

33 Player A submitted the highest bid and thus bought the good. She has to pay a price equal to her bid, namely Due to the risk she has to gamble at the end of the experiment (in case this auction is drawn to be payoff-relevant). Let s assume she is rolling a two with the dice. Hence, her valuation for the purchased good is reduced by 3. This results in the following earnings for this auction: A: (E) (B) (V) (R) = 7.70; B: (E); C: (E) In case this auction is selected to determine payoffs, players B and C receive 10.00, and player receives A b) Let s assume players submit the same bids as in a) but this time player A rolls a four at the end of the experiment. Hence, her valuation for the purchased good is increased by 3. This results in the following earnings in this auction: A: (E) (B) (V) (R) = 13.70; B: (E); C: (E) In case this auction is selected to determine payoffs, players B and C receive 10.00, and player receives A ) Questions: Please choose True or False : A player, who did not purchase a good, has zero earnings: True False A player who bids exactly her valuation for a non-risky good will earn 10 at most. True False A player who bids more than her valuation for a non-risky good and wins the auction will earn less than in case of not bidding at all. True False Altogether, you can t make losses in this part. True False If I submit a bid below my own valuation, I will earn 10 in case of not winning and the difference between my valuation and my bid in case of winning. True False The lower my bid, the lower my chance of winning the auction. True False The higher my bid, the higher my earnings in case of winning. True False 2) Exercises Players A, B and C have been grouped together. For a non-risky good they receive the following valuations: A: 5.50; B: 2.70; C: 5.60 Knowing their valuations the players submit the following bids: 33

34 A: 3.00; B: 2.00; C: 1.00 Which player purchases the good? Your answer: What are the earnings of player A? Your answer: What are the earnings of player B? Your answer: What are the earnings of player C? Your answer: Part 1 19 In this part you have to go through a number of lists. You can always choose between two alternatives in these lists: with option X you receive a lottery, and with option Y you receive a sure payment. On a given list, option X always represents the same lottery. Sure payments of option Y vary from decision to decision. Such a choice list could look as follows: 19 For the control experiment, i.e, the choice list. 34